In data sciences, OT distances can be used to perform various learning tasks over histograms (often called “bag of features”), such as image retrieval [1]}, [2]}.
OT distances are also at the heart of statistical tests [3]}, [4]}, [5]}, to check equality of two distributions from their samples, and to check the independence by comparing a joint distribution to the product of the marginals.
More advanced uses of OT consist in using it as a loss function for imaging and learning.
In supervised learning, the features and the output of the learned function are histograms compared using the OT loss. This can for instance be used for supervised vision tasks [6]}. This is also used for graph predictions [7]}, [8]}, where the features of all nodes on a graph are aggregated into a distribution in \(^d\) , and the OT distance serves to train an embedding of these node distributions modeled via a graph neural network [9]}.
In unsupervised learning, for density fitting, one optimizes some parametric model \(_\)  to fit empirical observations viewed as a discrete measure \(_N\) . This is achieved by minimizing \(\mapsto (_N,_)\) . A typical instantiation of this problem is to train a generative model [10]}, in which case the model is of the form \(_= ()_{\sharp } \beta _0\)  where \(\)  is a deep network and \(\beta _0\)  a fixed reference measure in a low dimensional latent space.
For imaging sciences, such as in shape matching (which is similar in spirit to generative model training), the transport map cannot be directly used for registration (because it lacks regularity).
Though the OT loss can be used to help the trained diffeomorphic models to avoid local minima of the registration energy  [11]}.
Other applications in imaging include inverse problems resolution, such as seismic tomography [12]} where the OT loss tends to reduce the presence of local minimizers when resolving the location of reflecting layers in the ground.
